Field of the Invention
The present invention relates in general to the field of electronics, and more specifically to a system and method for line cycle correlated spectral analysis for power measurement systems.
Description of the Related Art
Utility companies and other companies provide electrical power to many customers. The particular loads that utilize the electrical power can affect the quality of the delivered power. Total harmonic distortion (“THD”) represents one measure of the quality of the delivered power. Utility power transmission equipment and some loads are sensitive to the THD and non-linear loads typically increase the THD. THD is a line quality metric. Power is typically delivered with a line voltage having a fundamental line frequency. For example, the fundamental line frequency of the line voltage is 60 Hz in the United States of America and is 50 Hz in Europe. One way to calculate the THD of a signal is by taking the ratio of the total root mean square (“RMS”) of the signal at frequencies other than the fundamental line frequency to the total RMS. Equation 1 represents one measure of the THD in terms of a current component of the electrical power delivered to the load, and Equation 2 represents one measure of the THD in terms of a voltage component of the electrical power delivered to the load:
                              THD          I                =                              I                          H              ⁢              _              ⁢              RMS                                            I            RMS                                              Equation        ⁢                                  ⁢        1            
                              THD          V                =                              V                          H              ⁢              _              ⁢              RMS                                            V            RMS                                              Equation        ⁢                                  ⁢        2            In Equation 1, IH_RMS represents the root mean square of the current at harmonic frequencies of the fundamental line frequency, and IRMS represents the total root mean square of the current (which includes the fundamental line frequency and all harmonic frequencies). As previously stated, the fundamental line frequency is 60 Hz for the United States of America and 50 Hz for Europe. In Equation 2, VH_RMS represents the root mean square of the line voltage at harmonic frequencies of the fundamental line frequency, and VRMS represents the total root mean square of the voltage (which includes the fundamental line frequency and all harmonic frequencies).
Equation 3 represents a calculation of IH_RMS, and Equation 4 represents a calculation of VH_RMS:IH_RMS=√{square root over (IRMS2−IF_RMS2)}  Equation 3VH_RMS=√{square root over (VRMS2−VF_RMS2)}  Equation 4In Equation 3, IH_RMS represents the root mean square of the current at harmonic frequencies of the fundamental line frequency, IRMS2 is the square of total root mean square of the current delivered to the load, and IF_RMS is the square of root mean square of the current delivered to the load at the fundamental line frequency. Similarly in Equation 4, VH_RMS represents the root mean square of the line voltage at harmonic frequencies of the fundamental line frequency, VRMS2 is the square of total root mean square of the voltage delivered to the load, and VF_RMS2 is the root mean square of the voltage delivered to the load at the fundamental line frequency.
FIG. 1 depicts a power distribution and measurement system 100 that includes a voltage source 102 that provides a supply input voltage V and a current I to a load 104. The voltage V is, for example, a nominally 60 Hz/110 V line voltage in the United States of America or a nominally 50 Hz/220 V line voltage in Europe. The power distribution and measurement system 100 also includes a power measurement system 106 to determine the THD of the power delivered to the load 104. Normally the input voltage V is well regulated, so the THD in terms of the voltage is relatively small. However, the THD of the current I can vary significantly depending upon the load 104. Thus, a power measurement system 104 senses the input current I and determines the total harmonic distortion of the power delivered to the load 104 in terms of the input current I as, for example, defined in Equation 1.
The power measurement system 106 includes a THD processor 108 to determine the THD of the power delivered to the load 104 in terms of the current. The THD processor 108 is a digital signal processor and, thus, operates on samples I(n) of the current I in the discrete time domain, where “n” is a discrete index value. The THD processor 108 performs a spectral analysis of the sampled current I(n). Several algorithms exist to perform the spectral analysis. The Goertzel algorithm is one of the most computationally efficient digital signal processing technique for evaluating power at individual frequencies. Tn at least one embodiment, the THD processor 108 utilizes the Discrete Fourier Transform (DFT) algorithm to determine the THD. Equation 5 represents the DFT in the discrete time domain:
                              X          ⁡                      (            m            )                          =                              ∑                          n              =              0                                      N              -              1                                ⁢                                    x              ⁡                              (                n                )                                      ⁢                          e                                                -                  j2                                ⁢                                                                  ⁢                π                ⁢                                                                  ⁢                m                ⁢                                                                  ⁢                                  n                  /                  N                                                                                        Equation        ⁢                                  ⁢        5            
In Equation 5, X(m) represents the mth output frequency coefficient of some discrete time signal x(n), over the sample interval n=0 to n=N−1. For example, x(n) can be chosen to represent the samples of input current I. Then X(m) would represent the mth spectral component of current I. This mth spectral component is centered at 2πm/N, wherein N is the total number of samples used to compute the THD. For the fundamental spectral component, m equals N times the line frequency FL divided by the sampling frequency FS, i.e. m=N·FL/FS. Equation 6 represents the calculation of IF_RMS2 for Equation 3 in terms of Equation 5, and Equation 7 represents the calculation of IRMS2. The value IRMS is the total RMS of the current I, which includes the RMS of the fundamental frequency and all harmonics of the fundamental frequency for the current I. IF_RMS is the RMS of the fundamental frequency only. FL is the fundamental line frequency and FS is the sampling frequency of the input current I.
                                          I                          F              ⁢              _              ⁢              RMS                        2                    =                                                                      x                  ⁡                                      (                    m                    )                                                  2                                                              (                                      N                    /                    2                                    )                                2                                      2                          ⁢                                  ⁢                              Where            ⁢                                                  ⁢            m                    =                                                    F                L                                            F                S                                      *            N                                              Equation        ⁢                                  ⁢        6            
                              I          RMS          2                =                              1            N                    ⁢                                    ∑                              n                =                0                                            n                =                                  N                  -                  1                                                      ⁢                                          I                ⁡                                  (                  n                  )                                            2                                                          Equation        ⁢                                  ⁢        7            Determination of the THD in terms of the voltage V is identical to determination in terms of the current I except that x(n) represents samples of the voltage V.
The THD processor 108 then makes the determination of the THD available by transmitting the THD determination via transmitter 110, via display 112, and/or stored in memory 114 for subsequent access.
FIG. 2 depicts waveforms 200 of an exemplary line input current I and sample clock signal FS_CLK of the power distribution and measurement system 100. The clock signal FS_CLK is depicted by a series of pulses 202 that are exaggerated in time for illustration purposes. The input current I is periodic with a fundamental line frequency of FL. Referring to FIGS. 1 and 2, as previously stated, the THD processor 108 utilizes a fixed value of N samples for determination of the THD. However, the fundamental line frequency FL is not always constant and can slightly vary, such as between 59.99 Hz to 60.01 Hz. The frequency variation means that the number of samples N will not always represent an exact periodic waveform. For example, in the waveforms 200, the fundamental line frequency FL is slightly less than 60 Hz. Accordingly, as depicted within the dashed circle 204, at least the Nth sample of the input current I captures the beginning of a next period. When the N samples of the input current I do not represent a periodic waveform, spectral leakage occurs in the determination of the THD, and the spectral leakage represents an error in the calculation of the THD. Spectral leakage arises due to discontinuities at the end points of data sequences. Spectral measurements ideally require exact periodicity of the sampled input signal. In order to minimize spectral leakage, N should be chosen so that there is an integer number of cycles in the input data sequence of the frequency for which the THD is calculated. Typically, several windowing techniques are used to minimize discontinuity or the error from non-integer periodicity of the sampling clock and input signal within N samples. However, utilizing windows requires a significant amount of calculations that are expensive in terms of implementation in circuitry and expensive in terms of power utilized for each THD calculation. Furthermore, the fixed value of N results in spectral leakage when the line frequency of the current I varies.